Haspot, B;
Zatorska, E;
(2016)
From the highly compressible Navier-Stokes equations to the porous medium equation - Rate of convergence.
Discrete and Continuous Dynamical Systems- Series A
, 36
(6)
pp. 3107-3123.
10.3934/dcds.2016.36.3107.
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Abstract
We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to ε -1/2 for ε going to 0. When the initial velocity is related to the gradient of the initial density, the densities solving the compressible Navier-Stokes equations -ρ ε converge to the unique solution to the porous medium equation [14, 13]. For viscosity coefficient μ(ρ ε ) = ρ ε α with α > 1, we obtain a rate of convergence of ρ ε in L ∞ (0,T;H -1 (ℝ)); for 1 < α ≤ 3/2 the solution ρ ε converges in L ∞ (0,T;L 2 (ℝ)). For compactly supported initial data, we prove that most of the mass corresponding to solution ρ ε is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of ε.
Type: | Article |
---|---|
Title: | From the highly compressible Navier-Stokes equations to the porous medium equation - Rate of convergence |
Open access status: | An open access version is available from UCL Discovery |
DOI: | 10.3934/dcds.2016.36.3107 |
Publisher version: | http://dx.doi.org/10.3934/dcds.2016.36.3107 |
Language: | English |
Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
Keywords: | Compressible Navier-Stokes equations, porous medium equation, large Mach number |
UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
URI: | https://discovery.ucl.ac.uk/id/eprint/10035985 |
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